For many natural phenomena, models have a mathematical description in terms of coupled differential equations. When the equations are nonlinear, complex behaviour such as ‘chaos’ may result. An example is the system of three ordinary differential equations describing atmospheric convection developed by Edward Lorenz (1963). Power grids provide another case of complex nonlinear dynamics. Other examples are provided by numerical simulations in astrophysics or molecular dynamics. Solving these sets of nonlinear equations by analytical techniques can only be done rarely, so that numerical techniques are required. The equations may also have random terms to take account of noise in the system.
Conversely, one may ask the question whether the underlying dynamical model can be (partially) reconstructed from observations of the state of the system. This is called system identification. An example is Takens time series reconstruction, which applies nonlinear time series analysis of a sequence of observations of system over a long time interval. The complexity increases considerably in high dimensions, as in highly connective networks of systems. Important questions center around extreme events in dynamical systems, such as weather, climate, or ecological systems (‘tipping points’), or rare events such as nucleation events during phase transitions, conformal changes of molecules, and chemical reactions.
Observations can also be combined with numerical models in a data assimilation process, as in weather forecasting. Here observations are combined with the numerical results (the forecast) to obtain an estimate of the current state of the system. The model is then advanced in time and its result becomes the forecast in the next cycle.
- Can we extend the time series reconstruction method to the big data range?
- How can we obtain data-driven adaptive models for evolving (non-stationary) systems which are continuously monitored?
- Can we model the tails of distributions to distinguish between outliers (extreme events) and noise?Can we develop tools using big data sets to understand the dynamics and mechanisms leading to rare but important events in complex dynamical systems?
- Can we develop new algorithms for multiscale modeling that benefit from the parallel architecture of current computers and utilize accelerators?
|Laatst gewijzigd:||31 januari 2017 22:51|